Question: All of the 3rd grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$7.00$ each for teachers and $$2.50$ each for students, and the group paid $$43.50$ in total. The next month, the same group visited a science museum where the tickets cost $$28.00$ each for teachers and $$10.50$ each for students, and the group paid $$178.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+2.5y = 43.5}$ ${28x+10.5y = 178.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-28x-10y = -174}$ ${28x+10.5y = 178.5}$ Add the top and bottom equations together. $ 0.5y = 4.5 $ $ y = \dfrac{4.5}{0.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {7x+2.5y = 43.5}$ to find $x$ ${7x + 2.5}{(9)}{= 43.5}$ $7x+22.5 = 43.5$ $7x = 21$ $x = \dfrac{21}{7}$ ${x = 3}$ You can also plug ${y = 9}$ into $ {28x+10.5y = 178.5}$ and get the same answer for $x$ ${28x + 10.5}{(9)}{= 178.5}$ ${x = 3}$ There were $3$ teachers and $9$ students on the field trips.